William W. Tait

Infinitely Long Terms of Transfinite Type

Publisher Summary This chapter describes infinitely long terms of transfinite type. Functionals of higher type have been introduced into proof theory, which gives an interpretation of first order number theory in terms of the impredicative primitive recursive (p.r.) functionals of finite type. The chapter shows that for the consistency of number theory, Gentzens use of induction up to ɛ0 (with respect to p.r. properties) can be replaced by a quite different constructive.— but like Gentzens non-finitist - principle—namely, the assumption of constructive functionals of finite type and of their closure under p.r. operations. The chapter discusses only p.r. functionals or rather a certain generalization of them. The p.r. functionals of finite type can be generated from such φ by means of λ-abstraction and explicit definition.

Functionals Defined by Transfinite Recursion

This paper deals mainly with quantifier-free second order systems (i.e., with free variables for numbers and functions, and constants for numbers, functions, and functionals) whose basic rules are those of primitive recursive arithmetic together with definition of functionals by primitive recursion and explicit definition. Precise descriptions are given in §2. The additional rules have the form of definition by transfinite recursion up to some ordinal ξ (where ξ is represented by a primitive recursive (p.r.) ordering). In §3 we discuss some elementary closure properties (under rules of inference and definition) of systems with recursion up to ξ. Let Rξ denote (temporarily) the system with recursion up to ξ. The main results of this paper are of two sorts:Sections 5–7 are concerned with less elementary closure properties of the systems Rξ. Namely, we show that certain classes of functional equations in Rη can be solved in Rη for some explicitly determined η ξ). The classes of functional equations considered all have roughly the form of definition by recursion on the partial ordering of unsecured sequences of a given functional F, or on some ordering which is obtained from this by simple ordinal operations. The key lemma (Theorem 1) needed for the reduction of these equations to transfinite recursion is simply a sharpening of the Brouwer-Kleene idea.

A Counterexample to a Conjecture of Scott and Suppes

In [1], it is conjectured that if S is a sentence in the first-order functional calculus with identity, and every subsystem of every finite relational system which satisfies S also satisfies S , then S is finitely equivalent to a universal sentence. (Two sentences are finitely equivalent if and only if they are satisfied by the same finite relational systems.) The following sentence S refutes that conjecture, and moreover S is satisfied by all finite subsystems of all (finite or infinite) relational systems which satisfy it. 1 S contains as predicate letters only the two-place predicate letters ≦, R (and the identity symbol =).

Early analytic philosophy : Frege, Russell, Wittgenstein

These essays present new analyses of the central figures of analytic philosophy -- Frege, Russell, Moore, Wittgenstein, and Carnap -- from the beginnings of the analytic movement into the 1930s. The papers do not reflect a single perspective, but rather express divergent interpretations of this controversial intellectual milieu.

Gentzen’s Original Consistency Proof and the Bar Theorem

The story of Gentzen’s original consistency proof for first-order number theory [9], as told by Paul Bernays [1, 9], [11, Letter 69, pp. 76–79], is now familiar: Gentzen sent it off to Mathematische Annalen in August of 1935 and then withdrew it in December after receiving criticism and, in particular, the criticism that the proof used the Fan Theorem, a criticism that, as the references just cited seem to indicate, Bernays endorsed or initiated at the time but later rejected. That particular criticism is transparently false, but the argument of the paper remains nevertheless invalid from a constructive standpoint. In a letter to Bernays dated November 4, 1935, Gentzen protested this evaluation; but then, in another letter to him dated December 11, 1935, he admits that the ‘critical inference in my consistency proof is defective’.

Primitive Recursive Arithmetic and Its Role in the Foundations of Arithmetic: Historical and Philosophical Reflections

We discuss both the historical roots of Skolem’s primitive recursive arithmetic, its essential role in the foundations of arithmetic, its relation to the finitism of Hilbert and Bernays, and its relation to Kant’s philosophy of mathematics.

Proof-theoretic Semantics for Classical Mathematics

We discuss the semantical categories of base and object implicit in the Curry-Howard theory of types and we derive derive logic and, in particular, the comprehension principle in the classical version of the theory. Two results that apply to both the classical and the constructive theory are discussed. First, compositional semantics for the theory does not demand ‘incomplete objects’ in the sense of Frege: bound variables are in principle eliminable. Secondly, the relation of extensional equality for each type is definable in the Curry-Howard theory.

The Myth of the Mind

Of course, I do not mean by the title of this paper to deny the existence of something called ‘the mind’. But I do mean to call into question appeals to it in analyzing cognitive notions such as understanding and knowing, where its domain is taken to be independent of what one might find out in cognitive science. In this respect, I am expressing the skepticism of Sellars in “Empiricism and the philosophy of mind” [1956], where he explodes, not only the ‘Myth of the Given’, but also, as part of that myth, theorizing about thoughts, intentions and the like, where such theorizing is regarded as something more than a nascent cognitive science, in which such entities enter as theoretical entities, in aid of accounting for our cognitive abilities. The myth is that these entities present themselves in consciousness, available to us by introspection—and, perhaps, a priori reasoning. But, even among authors who claim to embrace Sellars’ critique of the Myth of the Given, his message about the mind is ignored. As an example, I want to consider and disarm an influential line of thought, by John McDowell, which implicates the mind in the analysis of knowing and understanding, not in the legitimate sense of suggesting causal accounts of our cognitive abilities in terms of mental or physiological structures, but in the sense of claiming that these abilities are mental or essentially involve the mental in a way that escapes the net of cognitive science. The ground on which I stand in this discussion is one which I attribute to Wittgenstein in his Philosophical Investigations. Basically, the position

First-Order Logic Without Bound Variables: Compositional Semantics

A strict version of compositional semantics would have all composite meaningful expressions be of the form \(XY\), where \(X\) and \(Y\) are meaningful and the concatenation expresses application of a function (\(X\)) to an argument (\(Y\)). In proof theory, compositionality is violated because of bound variables both in formulas (quantification) and in deductions (introduction rules). Two applications of typed combinator theory are used to introduce a proof theory for first-order predicate logic without identity in which there are no bound variables.